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Week of...

Notes and Links

1

Sep 10

About This Class, Tuesday, Thursday

2

Sep 17

HW1, Tuesday, Thursday, HW1 Solutions

3

Sep 24

HW2, Tuesday, Class Photo, Thursday

4

Oct 1

HW3, Tuesday, Thursday

5

Oct 8

HW4, Tuesday, Thursday

6

Oct 15

Tuesday, Thursday

7

Oct 22

HW5, Tuesday, Term Test was on Thursday. HW5 Solutions

8

Oct 29

Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class

9

Nov 5

Tuesday, Thursday

10

Nov 12

MondayTuesday is UofT November break, HW7, Thursday

11

Nov 19

HW8, Tuesday,Thursday

12

Nov 26

HW9, Tuesday , Thursday

13

Dec 3

Tuesday UofT Fall Semester ends Wednesday

F

Dec 10

The Final Exam (time, place, style, office hours times)

Register of Good Deeds

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Problem. Find the rank the matrix
$A={\begin{pmatrix}0&2&4&2&2\\4&4&4&8&0\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}}$.
Solution. Using (invertible!) row/column operations we aim to bring $A$ to look as close as possible to an identity matrix:
Do

Get

1. Bring a $1$ to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by $1/4$.

${\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}}$

2. Add $(8)$ times the first row to the third row, in order to cancel the $8$ in position 31.

${\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&6&8&6&2\\6&3&2&9&1\end{pmatrix}}$

3. Likewise add $(6)$ times the first row to the fourth row, in order to cancel the $6$ in position 41.

${\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&6&8&6&2\\0&3&4&3&1\end{pmatrix}}$

4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling).

${\begin{pmatrix}1&0&0&0&0\\0&2&4&2&2\\0&6&8&6&2\\0&3&4&3&1\end{pmatrix}}$

5. Turn the 22 entry to a $1$ by multiplying the second row by $1/2$.

${\begin{pmatrix}1&0&0&0&0\\0&1&2&1&1\\0&6&8&6&2\\0&3&4&3&1\end{pmatrix}}$

6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" $1$ at position 22.

${\begin{pmatrix}1&0&0&0&0\\0&1&2&1&1\\0&0&4&0&8\\0&0&2&0&4\end{pmatrix}}$

7. Using three column operations clean the second row except the pivot.

${\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&4&0&8\\0&0&2&0&4\end{pmatrix}}$

8. Clean up the row and the column of the $4$ in position 33 by first multiplying the third row by $1/4$ and then performing the appropriate row and column transformations. Notice that by pure luck, the $4$ at position 45 of the matrix gets killed in action.

${\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\end{pmatrix}}$

Thus the rank of our matrix is 3.
Lecture notes scanned by Zetalda